Reduced norms of division algebras over complete discrete valuation fields of local-global type

摘要

Let F be a complete discrete valuation field whose residue field k is a global field of positive characteristic p. Let D be a central division F-algebra of p-power degree. We prove that the subgroup of F* consisting of reduced norms of D is exactly the kernel of the cup product map lambda is an element of F* bar right arrow (D) boolean OR (lambda) is an element of H-3 (F, Q(p)/Z(p )(2)), if either D is tamely ramified or of period p. This gives a p-torsion counterpart of a recent theorem of Parimala, Preeti and Suresh, where the same result is proved for division algebras of prime-to-p degree.